Complex quadratic Abelian gauge condition ($F[A]={\partial }_{\mu }{A}_{\mu }+i{A}_{\mu }{A}_{\mu }$)

The formal variation of this condition is $\delta F=({\partial }^2+2iA\cdot \partial )\alpha$. Thus its Faddeev-Popov ghost field operator can be chosen as ${\mathcal{O}}_A=-{\partial }^2-2iA\cdot \partial$. With $D_{\mu }={\partial }_{\mu }+i{A}_{\mu }$, one has $D^2={\partial }^2+2iA\cdot \partial +iF[A]$, so ${\mathcal{O}}_A=-D^2$ on the formal gauge slice. The reality obstruction for a complex Euclidean gauge condition prevents interpreting this slice as an ordinary real Euclidean gauge fixing without an additional complex-contour prescription.

A massless complex bosonic charged scalar field with $D_{\mu }={\partial }_{\mu }+i{A}_{\mu }$ contributes $[\det (-D^2){]}^{-1}$ to a Gaussian background-field integral. A Faddeev-Popov ghost field pair contributes $\det {\mathcal{O}}_A$. On the formal complex quadratic Abelian gauge condition, ${\mathcal{O}}_A=-D^2$, so the two determinants cancel and their effective-action contributions sum to zero. Use matching regulators, boundary conditions and removed zero modes. This is a formal cancellation of closed loops; the reality obstruction for a complex Euclidean gauge condition precludes deducing that physical scalar quantum electrodynamics has no quantum effects.

For real $A_{\mu }$ in positive-definite Euclidean signature, $\partial \cdot A$ is real and $A_{\mu }{A}_{\mu }\ge 0$. Therefore $\partial \cdot A+i{A}^2=0$ forces $A^2=0$, and hence $A=0$. Its only real gauge orbit is the zero-field orbit; a field with nonzero electromagnetic curvature cannot be gauge-transformed to zero. Consequently this complex equation is not an admissible gauge fixing for general Hermitian Euclidean U(1) gauge symmetry configurations. A formal complexified treatment needs a specified contour and cannot be justified by a real delta-functional insertion.